Banach Journal of Mathematical Analysis

Linear maps preserving pseudospectrum and condition spectrum

G. Krishna Kumar and S. H. Kulkarni

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We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

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Banach J. Math. Anal., Volume 6, Number 1 (2012), 45-60.

First available in Project Euclid: 14 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 46H05: General theory of topological algebras 46J05: General theory of commutative topological algebras 47S48

Pseudospectrum condition spectrum almost multiplicative map linear preserver perturbation


Krishna Kumar, G.; Kulkarni, S. H. Linear maps preserving pseudospectrum and condition spectrum. Banach J. Math. Anal. 6 (2012), no. 1, 45--60. doi:10.15352/bjma/1337014664.

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  • J. Alaminos, J. Extremera and A.R. Villena, Approximately spectrum-preserving maps, J. Funct. Anal. 261 (2011), no. 1, 233–266
  • Z. Bai and J. Hou, Characterizing isomorphisms between standard operator algebras by spectral functions, J. Operator Theory 54 (2005), no. 2, 291–303.
  • C.-K. Li and N.K. Tsing, Linear Presrrver Problems: A Brief Inroduction and Some Special Techniques, Linear Algebra Appl. 153 (1992), 217–235.
  • I. Gohberg, S. Goldberg and M.A. Kaashoek, Basic classes of linear operators, Birkhäuser, Basel, 2003.
  • R. Hagen, S. Roch and B. Silbermann, $C\sp *$-algebras and numerical analysis, Monographs and Textbooks in Pure and Applied Mathematics, 236, Dekker, New York, 2001.
  • P.R. Halmos, A Hilbert space problem book, Second edition, Springer, New York, 1982.
  • A.A. Jafarian and A.R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261.
  • K. Jarosz, Perturbations of Banach algebras, Lecture Notes in Math., 1120, Springer, Berlin, 1985.
  • J. Cui, V. Forstall, C.-K. Li and V. Yannello, Properties and preservers of the pseudospectrum, Linear Algebra Appl, (2011), doi:10.1016/j.laa.2011.03.044.
  • B.E. Johnson, Approximately multiplicative functionals, J. London Math. Soc. (2) 34 (1986), no. 3, 489–510.
  • B.E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), no. 2, 294–316.
  • S.H. Kulkarni and D. Sukumar, Almost multiplicative functions on commutative Banach algebras, Studia Math. 197 (2010), no. 1, 93–99.
  • S.H. Kulkarni and D. Sukumar, The condition spectrum, Acta Sci. Math. (Szeged) 74 (2008), no. 3-4, 625–641.
  • M. Marcus and B.N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math. 11 (1959), 61–66.
  • L. Molnár, Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Mathematics, 1895, Springer, Berlin, 2007.
  • M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kōdai Math. Sem. Rep. 11 (1959), 182–188.
  • T.W. Palmer, Banach algebras and the general theory of $\sp *$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, 49, Cambridge Univ. Press, Cambridge, 1994.
  • T.J. Ransford, Generalised spectra and analytic multivalued functions, J. London Math. Soc. (2) 29 (1984), no. 2, 306–322.
  • W. Rudin, Functional analysis, Second edition, McGraw-Hill, New York, 1991.
  • A.R. Sourour, Invertibility preserving linear maps on $ L(X)$, Trans. Amer. Math. Soc. 348 (1996), no. 1, 13–30.
  • L.N. Trefethen and M. Embree, Spectra and pseudospectra, Princeton Univ. Press, Princeton, NJ, 2005.
  • W. Żelazko, A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83–85.